Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches

Bresch, Didier; Fernández-Nieto, Enrique D.; Ionescu, Ioan R.; Vigneaux, Paul

doi:10.1007/978-3-0346-0152-8_4

Cited by 27 publications

(34 citation statements)

References 37 publications

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“…Note however that in these works nonconforming methods are used, and require further works for proving their convergence. Related works [5,17] use the augmented Lagrangian method.…”

Section: Resultsmentioning

confidence: 99%

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Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

2014

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We study approximations by conforming methods of the solution to the variational inequality ∂ t u, v − u + ψ(v) − ψ(u) ≥ f, v − u , which arises in the context of inviscid incompressible Bingham type fluid flows and of the total variation flow problem. In the general context of a convex lower semi-continuous functional ψ on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for non-smooth data. Then we introduce a general class of total variation functionals ψ, for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms, and prove their convergence for general total variation functionals. A comparison with an analytical solution concludes this study.

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“…Note however that in these works nonconforming methods are used, and require further works for proving their convergence. Related works [5,17] use the augmented Lagrangian method.…”

Section: Resultsmentioning

confidence: 99%

“…Assuming a constant density ρ = 1 for the fluid and neglecting the nonlinear convection term, the momentum conservation equation is given by 5) where f has values in R N . An initial condition…”

Section: Introductionmentioning

confidence: 99%

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

2014

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“…The derivation being very close, we refer to [8] for more details and here we directly present the resulting model. Its physical characteristics and relevance are briefly described in the following.…”

Section: Model and Resolution Approachesmentioning

confidence: 99%

Efficient numerical schemes for viscoplastic avalanches. Part 2: The 2D case

Fernández-Nieto

Gallardo

Vigneaux

2018

Journal of Computational Physics

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This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated to the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez-Moreno) to discretize the problem. To be able to accurately simulate the stopping behaviour of the avalanche, new schemes need to be designed, involving the classical notion of wellbalancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.

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“…As a model problem for viscoplastic shallow flows, we use the Bingham shallow-water model derived in [8], but we add here the fact that the bottom is more general than a plane slope. The derivation being very close, we refer to [8] for more details and here we directly present the resulting model.…”

Section: Model and Resolution Approachesmentioning

confidence: 99%

“…In this context, a careful treatment must be made to design a well-balanced scheme when coupling the finite volume scheme and the duality method. This idea was first introduced in [8], in the context of a Bingham fluid treated with an AL method. We extend here this idea for a Shallow-Water-Bingham model on a general topography and in the presence of wet/dry fronts.…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical schemes for viscoplastic avalanches. Part 1: The 1D case

Fernández-Nieto

Gallardo

Vigneaux

2014

Journal of Computational Physics

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This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated to the yield threshold: finitevolume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez-Moreno) to discretize the problem. To be able to accurately simulate the stopping behaviour of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.

show abstract

Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches

Cited by 27 publications

References 37 publications

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

Efficient numerical schemes for viscoplastic avalanches. Part 2: The 2D case

Efficient numerical schemes for viscoplastic avalanches. Part 1: The 1D case

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